Solutions of Lévy-driven SDEs with unbounded coefficients as Feller processes

• Published in: Proceedings of the American Mathematical Society 2018-08, Vol.146 (8), p.3591-3604
• Main Author: Kühn, Franziska
• Format: Article
• Language: English
• Subjects: F. STATISTICS AND PROBABILITY; Research article
• ISSN: 0002-9939; 1088-6826
• DOI: 10.1090/proc/14022

Let (Lt)t≥0(L_t)_{t \geq 0} be a kk-dimensional Lévy process and σ:Rd→Rd×k\sigma : \mathbb {R}^d \to \mathbb {R}^{d \times k} a continuous function such that the Lévy-driven stochastic differential equation (SDE) dXt=σ(Xt−)dLt,X0∼μ,\begin{equation*} dX_t = \sigma (X_{t-}) \, dL_t, \qquad X_0 \sim \mu , \end{equation*} has a unique weak solution. We show that the solution is a Feller process whose domain of the generator contains the smooth functions with compact support if and only if the Lévy measure ν\nu of the driving Lévy process (Lt)t≥0(L_t)_{t \geq 0} satisfies ν({y∈Rk;|σ(x)y+x|>r})→|x|→∞0.\begin{equation*} \nu (\{y \in \mathbb {R}^k; |\sigma (x)y+x|>r\}) \xrightarrow []{|x| \to \infty } 0. \end{equation*} This generalizes a result by Schilling & Schnurr (2010) which states that the solution to the SDE has this property if σ\sigma is bounded.